People while taking short trips have different priorities of what/where/how to cover a whole city. When it comes to Seoul - very populous and a lot of intersting things to see that are scattered all over the place then my list goes like this:
1- Seoul City Tour Bus: By taking this tour one can cover all the major toursit spots worth seeing (such as palaces, museums) with a hop on/off option for mere 10 USD or 10,000 won...there is a night tour option- which is worth a try!
2- Namsan Tower: Special environmentally friendly buses are run by the Seoul city government for only 1$ or 1000 won a ride...from here one can have a panoramic view of entire Seoul and can dine at their revolving restaurant and visit the mini bear museum.
3- Itaewon: From Namsan - just take a bus down to Itaewon and here we have hundreds of restaurants from all the different corners of the world such as Bulgarian, Serbian, Greek, Morroccan and countless Halal Indian/Pakistani restaurants as well as the Islamic Center of Korea and the only mosque of Seoul.
4- War Memorial of Korea: A 5 minutes drive from Itaewon between Noksapyeong and Samgakchi is this amazing War Memorial!
5- Namdaemun Market: Take a taxi from war memorial and it may only cost you 3$ or 3,000 won and a 10 minutes drives to get to one of the 'Traditional Markets' of Seoul...here you can buy almost everything and much cheaper than at any other place!
6- Myeong Dong: Take a Bus from exit 5 of Namdaemun and get off at Myeong Dong - famous place for baragains and great deals ranging from fashion accessories, clothes, shoes and bags...it also has a good number of restaurants!
7- Cheonggye[chun] Stream (Gwanghwamun Area): From Myeong Dong take subway and get off at Gwanghwamun Station - exit:5...you will be in front of KT building - in and around the vicinity are the Lee Sungshin's statue and also King Sejong's statue, This is Sejongno...where we have the Duty Free shops, Sejong Arts Center, Cheongyyechun Stream, Gyeongbok Palace, major government offices as Ministry of Tourism and Culture and Ministry of Foreign Affairs and Trade.
9- Insadong Street/Jogyesa Temple.: This is a 'must see' spot - a small street filled with coffee shops, cafes, restaurants, art gallaries, souvenirs (the best ones in Korea). Just around the corner is Jogyesa- one of the main temples of Jogye Buddhism and its head quarter. With a few hundred years of history and right in the heart of the city- one must not miss it. This is also the main venue of "Buddah's Birthday" celebrations.
10- Bukchon: Tradional Hanoks or Korean houses are here. This neighborhood in Joseon Dynasty was the residential area for ministers or the yangban (nobles) of the time etc.
11- Gangnam/COEX: One place that one must visit is the Gangnam area..both sides of the street has very intersting shops and activities and loads of young people...just nearby, if you get off at Samsong Station and take exit 5...you will be at COEX...a lot of shopping malls, CGV theater, open air concerts and just around the corner is another temple: Bongeungsa...do visit it...this is a mellange of traditional and modern architectural site!
12- Yeouido Island/Ferry Ride: Another important site...its importance is the same as the Wall Street- New York - with Korea's Stock Exchange and the tallest building in Korea 63 Building...are the main attractions..one can also take a ferry ride from here to explore the Han River for just 7,000 won or 7$!